Nanomanufacturing is critical to the future growth of U.S. manufacturing. Yet the process yield of current nanodevices is typically 10% or less. Particularly in nanomaterials growth, there may exist large variability across the sites on a substrate, which could lead to variability in properties. Essential to the reduction of variability is to mathematically describe the spatial variation of nanostructure. This research therefore aims at a method of modeling and estimating nanostructure morphology variation for process yield improvement. This method consists of (1) morphology variation modeling based on Gaussian Markov random field (GMRF) theory, and (2) maximum likelihood estimation (MLE) of morphology variation model based on measurement data. The research challenge lies in the proper definition and estimation of the interactions among neighboring nanostructures. To model morphology variation, nanostructures on all sites are collectively described as a GMRF. The morphology variation model serves for the space-time growth model of nanostructures. The probability structure of the GMRF is specified by a so-called simultaneous autoregressive scheme, which defines the neighborhood systems for any site on a substrate. The neighborhood system characterizes the interactions among adjacent nanostructures by determining neighbors and their influence on a given site in terms of conditional auto-regression. The conditional auto-regression representation uniquely determines the precision matrix of the GMRF. Simulation of nanostructure morphology variation is conducted for various neighborhood structures. Considering the boundary effects, both finite lattice and infinite lattice models are discussed. The simultaneous autoregressive scheme of the GMRF is estimated via the maximum likelihood estimation (MLE) method. The MLE estimation of morphology variation requires the approximation of the determinant of the precision matrix in the GMRF. The absolute term in the double Fourier expansion of a determinant function is used to approximate the coefficients in the precision matrix. Since the conditional MLE estimates of the parameters are affected by coding the date, different coding schemes are considered in the estimation based on numerical simulation and the data collected from SEM images. The results show that the nanostructure morphology variation modeling and estimation method could provide tools for yield improvement in nanomanufacturing.

Les inondations constituent le risque naturel prédominant dans le monde et les dégâts qu'elles causent sont les plus importants parmi les catastrophes naturelles. Un des principaux facteurs expliquant les inondations sont les précipitations extrêmes. En raison des changements climatiques, l'occurrence et l'intensité de ces dernières risquent fort probablement de s'accroître. Par conséquent, le risque d'inondation pourrait vraisemblablement s'intensifier. Les impacts de l'évolution des précipitations extrêmes sont désormais un enjeu important pour la sécurité du public et pour la pérennité des infrastructures. Les stratégies de gestion du risque d'inondation dans le climat futur sont essentiellement basées sur les simulations provenant des modèles numériques de climat. Un modèle numérique de climat procure notamment une série chronologique des précipitations pour chacun des points de grille composant son domaine spatial de simulation. Les séries chronologiques simulées peuvent être journalières ou infra-journalières et elles s'étendent sur toute la période de simulation, typiquement entre 1961 et 2100. La continuité spatiale des processus physiques simulés induit une cohérence spatiale parmi les séries chronologiques. Autrement dit, les séries chronologiques provenant de points de grille avoisinants partagent souvent des caractéristiques semblables. De façon générale, la théorie des valeurs extrêmes est appliquée à ces séries chronologiques simulées pour estimer les quantiles correspondants à un certain niveau de risque. La plupart du temps, la variance d'estimation est considérable en raison du nombre limité de précipitations extrêmes disponibles et celle-ci peut jouer un rôle déterminant dans l'élaboration des stratégies de gestion du risque. Par conséquent, un modèle statistique permettant d'estimer de façon précise les quantiles de précipitations extrêmes simulées par un modèle numérique de climat a été développé dans cette thèse. Le modèle développé est spécialement adapté aux données générées par un modèle de climat. En particulier, il exploite l'information contenue dans les séries journalières continues pour améliorer l'estimation des quantiles non stationnaires et ce, sans effectuer d'hypothèse contraignante sur la nature de la non-stationnarité. Le modèle exploite également l'information contenue dans la cohérence spatiale des précipitations extrêmes. Celle-ci est modélisée par un modèle hiérarchique bayésien où les lois a priori des paramètres sont des processus spatiaux, en l'occurrence des champs de Markov gaussiens. L'application du modèle développé à une simulation générée par le Modèle régional canadien du climat a permis de réduire considérablement la variance d'estimation des quantiles en Amérique du Nord. / Precipitation extremes plays a major role in flooding events and their occurrence as well as their intensity are expected to increase. It is therefore important to anticipate the impacts of such an increase to ensure the public safety and the infrastructure sustainability. Since climate models are the only tools for providing quantitative projections of precipitation, flood risk management for the future climate may be based on their simulations. Most of the time, the Extreme value theory is used to estimate the extreme precipitations from a climate simulation, such as the T-year return levels. The variance of the estimations are generally large notably because the sample size of the maxima series are short. Such variance could have a significant impact for flood risk management. It is therefore relevant to reduce the estimation variance of simulated return levels. For this purpose, the aim of this paper is to develop a non-stationary and regional statistical model especially suited for climate models that estimates precipitation extremes. At first, the non-stationarity is removed by a preprocessing approach. Thereafter, the spatial correlation is modeled by a Bayesian hierarchical model including an intrinsic Gaussian Markov random field. The model has been used to estimate the 100-year return levels over North America from a simulation by the Canadian Regional Climate Model. The results show a large estimation variance reduction when using the regional model.

Estimating large covariance matrices from small samples is an important problem in many fields. Among others, this includes spatial statistics and data assimilation. In this thesis, we deal with several methods of covariance estimation with emphasis on regula- rization and covariance models useful in filtering problems. We prove several properties of estimators and propose a new filtering method. After a brief summary of basic esti- mating methods used in data assimilation, the attention is shifted to covariance models. We show a distinct type of hierarchy in nested models applied to the spectral diagonal covariance matrix: explicit estimators of parameters are computed by the maximum like- lihood method and asymptotic variance of these estimators is shown to decrease when the maximization is restricted to a subspace that contains the true parameter value. A similar result is obtained for general M-estimators. For more complex covariance mo- dels, maximum likelihood method cannot provide explicit parameter estimates. In the case of a linear model for a precision matrix, however, consistent estimator in a closed form can be computed by the score matching method. Modelling of the precision ma- trix is particularly beneficial in Gaussian Markov random fields (GMRF), which possess a sparse precision matrix. The...

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A..=2b, where A 2 Rnn is a large, sparse symmetric positive definite matrix and b 2 Rn is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LLT is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L..T z, with x = A..1=2z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form n = A..=2b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t..=2 and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A..=2b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.